Stress Invariants

For a concrete model to be most useful, the model itself should be defined independent

of the coordinate system attached to the material. Thus, it is necessary to define

the model in terms of stress invariants which are, by definition, independent of the

coordinate system selected. The three-dimensional stress state of the material is

traditionally defined by the stress tensor, which can be represented relative to a chosen

coordinate system by a matrix:

This stress tensor is often decomposed into two parts: a purely hydrostatic stress, σm,

defined in Equation 2.2, and the deviatoric stress tensor, sij , defined in Equation 2.3.

A common set of stress invariants are the three principal stress invariants. The

principal stress coordinate system is the coordinate system in which shear stresses

vanish, leaving only normal stresses. This requirement of zero shear stresses leads to

the characteristic equation:

The first, second, and third invariant of the stress tensor, I1, I2, and I3 are defined

in the following equations:

The three roots of Equation 2.4 are the three principal stress invariants, also called

the three principal stresses. They are ordered so that σ1 > σ2 > σ3.

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The three principal stresses, as well as most other stress invariants, can be rewritten

in terms of three core invariants: the first invariant of the stress tensor, I1, and

the second and third invariants of the deviatoric stress tensor, J2 and J3. The first

invariant of the stress tensor, I1, was previously defined in Equation 2.5. The second

and third invariants of the deviatoric stress tensor are defined as:

Clearly, a large variety of stress invariants were available to use in defining the

model. The three stress invariants _, r, and _ were chosen to define the components

of the concrete model:

They have a direct physical interpretation which makes it easier to understand the

physical implications of the model. To understand the physical significance of each

of these invariants, it is helpful to look at them in the principal stress coordinate

system (σ1, σ2, σ3). Recall that the principal stress coordinate system corresponds

to the orientation in which the material has no shear stresses. A diagram of this

coordinate system is shown in Figure 2.1. Consider the case of purely hydrostatic

loading with magnitude equal to σh. For this load case, σ1 = σ2 = σ3 = σh. Thus,

the load path travels along the ξ axis. The magnitude of the hydrostatic load, σh, is

equal to the stress invariant ξ. Therefore, it is clear that the invariant ξ represents the

hydrostatic component of the current stress state. Now we consider the planes that lie

perpendicular to this hydrostatic axis. For any given stress state lying in one of these

planes, the distance between the point representing the stress state in the principal

stress coordinate system and the hydrostatic axis is related to the deviatoric stress.

The magnitude of this distance is equal to the invariant r. Thus, r represents the

stress invariant measure of the deviatoric stress. This leaves only the third invariant,

θ, also known as the Lode angle. The invariant θ is controlled by the relationship

of the intermediate principal stress to the major and minor principal stresses. When

the intermediate principal stress, σ2, is equal to the minor principal stress, σ3, the

value for θ becomes 600. When the intermediate principal stress, σ2, is equal to the

major principal stress, σ1, the value for θ becomes 00. Thus, θ is an indication of

the magnitude of the intermediate principal stress in relation to the minor and major

principal stresses.

Thick walled cylinders are widely used in chemical, petroleum, military industries as well as in nuclear power plants. They are usually subjected to high pressure & temperatures which may be constant or cycling. Industrial problems often witness ductile fracture of materials due to some discontinuity in geometry or material characteristics. The conventional elastic analysis of thick walled cylinders to final radial & hoop stresses is applicable for the internal pressure up to yield strength of material.

General applicationn of Thick- Walled cylinders include, high pressure reactor vessels used in mettalurgical operations, process plants, air compressor units, pneumatic reservoirs, hydraulic tanks, storage for gases like butane LPG etc.

In this Project we are going to analyze effect of internal and External Pressure on Thick walled cylinder , How radial stress & hoop Stress will vary with change of radius. Contact pressure in shrink Fit and it’s affect on hoop stress and radial stress in analysed.